I just read a study which surprised me, and then everything made sense.
A psychologist took a group of people, and assessed their cognitive skills and their mood after a normal nights' sleep. They then either sleep deprived these people by only letting them sleep for four hours for the entire night OR they let them sleep eight hours, but woke them up four times during the night and made them stay awake for 15 minutes each time.
Not surprisingly, after the sleep deprivation, people were much worse at paying attention, solving problems and reported that their mood was not good.
But the really interesting thing: there was NO significant difference between the people who had only been allowed to sleep four hours compared to the people who had eight hours with interruptions. Getting interrupted sleep was just as bad for mood and thinking as only getting a paltry four solid hours.
Are you being woken up at night by a child? It's no wonder you sometimes struggle the following day. Despite being asleep for an eight hour period, your interruptions mean you may as well have got four hours.
Another reason to be compassionate towards yourself if you feel tired and cranky!